Sequence and Series

Arithmetic Sequence

An arithmetic sequence (or arithmetic progression) is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference (d). 

Formula for the nth term of an arithmetic sequence: 

Where: 

- \( t_n \) = \(n^{th}\) term , \( a \) = first term , \( d \) = common difference , \( n \) = term number 

Example: 

Find the 10th term of the sequence: 2, 5, 8, 11, ... 

- Here, \( a = 2 \), \( d = 3 \), and \( n = 10 \). 

- Using the formula: 

  \( t_{10} = 2 + (10 - 1) \times 3 = 2 + 27 = 29\)

- Answer: The 10th term is 29. 

Means of an Arithmetic Sequence

The arithmetic means represented by \(m_1, m_2, m_3, ……. m_n\) are the values between the first and the last terms in an arithmetic sequence. 

Formula for the arithmetic mean:

1. If there are two first(a) and last term(b): \(a, m, b\)

To find the a mean value \(m\) between the first and last term, we generally calculate arithmetic mean. Here is the formula and example.

\(M = \frac{a + b}{2}\) 

Where \( a \) and \( b \) are two terms in the sequence. 

Example: 

Q. Find the arithmetic mean between 4 and 10. 

- Here, the sequence is, 4, m, 10

\(M = \frac{4 + 10}{2} = \frac{14}{2} = 7\) 

- Answer: The arithmetic mean is 7. 

2. If there are many means i.e. \(m_n\): \(a, m_1, m_2, …. M_n, b\)

To find more than one means we first calculate the common difference (d) from the last term \(t_n / b\). Then, we simply use following formulas to calculate \(m_1, m_2 …..\).

Sum of an Arithmetic Series

The sum of the first n terms of an arithmetic sequence is called an arithmetic series. It is represented by \(S_n\).

Formula for the sum of an arithmetic series: 

1. If first term (a), common difference (d) are given:

\(S_n = \frac{n}{2} [2a_1 + (n - 1) d]\)

2. If first term (a) and last term \(t_n\) are given:

\(S_n = \frac{n}{2} (a_1 + a_n)\) 

Example: 

Find the sum of the first 10 terms of the sequence 3, 6, 9, 12, ... 

- Given: \( a_1 = 3 \), \( d = 3 \), \( n = 10 \). 

- Find the 10th term: 

  \( a_{10} = 3 + (10 - 1) \times 3 = 3 + 27 = 30\) 

- Calculate the sum: 

  \ S_{10} = \frac{10}{2} (3 + 30) = 5 \times 33 = 165\) 

- Answer: The sum is 165. 

Geometric Sequence

A geometric sequence (or geometric progression) is a sequence where each term is found by multiplying the previous term by a constant called the common ratio (r). 

Formula for the nth term of a geometric sequence: 

\(a_n = a_1 \cdot r^{(n-1)}\)

Example: 

Find the 6th term of the sequence: 3, 6, 12, 24, ... 

- Here, \( a_1 = 3 \), \( r = 2 \), and \( n = 6 \). 

- Using the formula: 

  \(a_6 = 3 \times 2^{(6-1)} = 3 \times 2^5 = 3 \times 32 = 96\) 

- Answer: The 6th term is 96. 

Means of a Geometric Sequence

Formula for geometric mean: 

1. If there is a mean between first term (a) and last term (b): \(a, m, b\)

The geometric mean of two numbers is the square root of their product. 

\(M = \sqrt{a \times b}\ 

Example: 

Find the geometric mean of 4 and 16. 

\(M = \sqrt{4 \times 16} = \sqrt{64} = 8\) 

- Answer: The geometric mean is 8. 

2. If there are many means i.e. \(m_n\): \(a, m_1, m_2, …. M_n, b\)

To find more than one means we first calculate the common ratio (r) from the last term \(t_n / b\). Then, we simply use following formulas to calculate \(m_1, m_2 …..\).

Sum of a Geometric Series

The formulas for calculating sum of the first n terms of a geometric sequence are given below.

1. If first term (a) and common ration (r) are given.

a. When r is greater than 1 i.e. \(r > 1\):

\(S_n = a \frac{(r^n - 1)}{r - 1}, \quad \text{for } r < 1\) 

b. When r is less than 1 i.e. \(r < 1\):

\(S_n = a \frac{(1 - r^n)}{1 - r}, \quad \text{for } r < 1\) 

2. If first term (a), common ratio (r) and last term \(t_n\) are given.

\(S_n = \frac{t_n r - a}{r - 1}\)

Example: 

Find the sum of the first 5 terms of the sequence: 2, 6, 18, 54, ... 

- Given: \( a_1 = 2 \), \( r = 3 \), \( n = 5 \). 

- Using the formula: 

  \( S_5 = 2 \frac{1 - 3^5}{1 - 3} = 2 \frac{1 - 243}{-2} = 2 \frac{-242}{-2} = 2 \times 121 = 242  \)

- Answer: The sum is 242. 

Key points for the chapter Sequence and Series 

- Arithmetic sequences have a constant difference. 

- Geometric sequences have a constant ratio. 

- Arithmetic means and geometric means help find averages between numbers. 

- Summation formulas allow finding total sums efficiently.